Logarithms Quick Reference Cheat Sheet

Key Facts

• The definition of a logarithm is $\log_b(x)=y$ if and only if $b^y=x$, where $b>0$, $b\ne 1$, and $x>0$.
• The product rule is $\log_b(MN)=\log_b(M)+\log_b(N)$ for positive $M$ and $N$.
• The quotient rule is $\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$ for positive $M$ and $N$.
• The power rule is $\log_b(M^p)=p\log_b(M)$ for positive $M$.
• The change of base formula is $\log_b(x)=\frac{\log_a(x)}{\log_a(b)}$, commonly written as $\log_b(x)=\frac{\log(x)}{\log(b)}$.
• Common logarithms use base $10$, so $\log(x)=\log_{10}(x)$.
• Natural logarithms use base $e$, so $\ln(x)=\log_e(x)$.
• Inverse identities include $\log_b(b^x)=x$ and $b^{\log_b(x)}=x$ for valid values of $b$ and $x$.

Vocabulary

Logarithm

A logarithm is the exponent needed to raise a base to a given positive number.

Base

The base is the positive number $b$ in $\log_b(x)$, where $b\ne 1$.

Argument

The argument is the input $x$ in $\log_b(x)$, and it must be positive.

Common Logarithm

A common logarithm is a logarithm with base $10$, written as $\log(x)$.

Natural Logarithm

A natural logarithm is a logarithm with base $e$, written as $\ln(x)$.

Change of Base

Change of base is the formula $\log_b(x)=\frac{\log_a(x)}{\log_a(b)}$ used to rewrite a logarithm in a different base.